I have a sum of two Gaussian type functions, $g_1(x) = C_1 Exp[-\alpha (X_1-x)^2]$ and $g_2(x) = C_2 Exp[-\beta (X_2-x)^2]$ and have found that the derivative w.r.t. $x$ is
$f(x) = 2 C_1 (X_1 - x) \alpha Exp[ -\alpha (X_2 - x)^2] + 2 C_2 (X_2 - x) \beta Exp[ -\beta (X_2 - x)^2]$
I would like to find where this function reaches a minimum and so I am trying to solve:
$C_1 (X_1 - x) \alpha Exp[ -\alpha (X_2 - x)^2] + C_2 (X_2 - x) \beta Exp[ -\beta (X_2 - x)^2] = 0$
for $x$. I thought this would be rather simple and just didn't want to work out the math by hand but Mathematica tells me that it cannot solve it. I thought this would be College Algebra level math involved here but it seems a bit more complicated than that now that I look at it. Any ideas?